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July  2012, 11(4): 1397-1406. doi: 10.3934/cpaa.2012.11.1397

Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity

Adrian Constantin 1,

1. 

University of Vienna, Fakultät für Mathematik, Nordbergstraße 15, 1090 Vienna

Received  April 2011 Revised  September 2011 Published  January 2012

In the absence of stagnation points, we derive the dispersion relation for periodic travelling waves of small amplitude propagating at the surface of water with a layer of constant vorticity adjacent to the flat bed within an irrotational flow.
Keywords: Water waves, discontinuous vorticity, dispersion relation..
Mathematics Subject Classification: Primary: 35Q31, 76D33, 34B05; Secondary: 12D1.
Citation: Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397
References:
[1]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[2]

A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16. doi: 10.1016/j.euromechflu.2010.09.008.  Google Scholar

[3]

A. Constantin, "Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis," CBMS-NSF Series in Applied Mathematics, Vol. 81, SIAM, Philadelphia, 2011. Google Scholar

[4]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.  Google Scholar

[5]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773.  Google Scholar

[6]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[7]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[8]

A. Constantin, J. Escher and H.-C. Hsu, Pressure beneath a solitary water wave: mathematical theory and experiments, Arch. Rational Mech. Anal., 201 (2011), 251-269. doi: 10.1007/s00205-011-0396-0.  Google Scholar

[9]

A. Constantin, D. Sattinger and W. Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech., 548 (2006), 151-163. doi: 10.1017/S0022112005007469.  Google Scholar

[10]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.  Google Scholar

[11]

A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950. doi: 10.1002/cpa.20165.  Google Scholar

[12]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Philos. Trans. Roy. Soc. London A, 365 (2007), 2227-2239. doi: 10.1098/rsta.2007.2004.  Google Scholar

[13]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.  Google Scholar

[14]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Rational Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4.  Google Scholar

[15]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Rational Mech. Anal., 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x.  Google Scholar

[16]

A. F. Teles da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), 281-302. doi: 10.1017/S0022112088002423.  Google Scholar

[17]

M. Ehrnström, On the streamlines and particle paths of gravitational water waves, Nonlinearity, 21 (2008), 1141-1154. doi: 10.1088/0951-7715/21/5/012.  Google Scholar

[18]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 2001.  Google Scholar

[19]

M. Goldshtik and F. Hussain, Inviscid separation in steady planar flows, Fluid Dynamics Research, 23 (1998), 235-266. doi: 10.1016/S0169-5983(98)00017-3.  Google Scholar

[20]

D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves, Philos. Trans. Roy. Soc. London A, 365 (2007), 2241-2251. doi: 10.1098/rsta.2007.2005.  Google Scholar

[21]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows,, Quart. Appl. Math. (in print)., ().   Google Scholar

[22]

I. G. Jonsson, Wave-current interactions, in "The Sea" (eds. B. Le Mehaute and D. M. Hanes), J. Wiley, (1990), 65-120. Google Scholar

[23]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215. doi: 10.1017/S0022112008002371.  Google Scholar

[24]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves, Eur. J. Mech. B Fluids, 27 (2008), 96-109. doi: 10.1016/j.euromechflu.2007.04.004.  Google Scholar

[25]

D. H. Peregrine, Interaction of water waves and currents, Adv. Appl. Mech., 16 (1976), 9-117. doi: 10.1016/S0065-2156(08)70087-5.  Google Scholar

[26]

V. V. Prasolov, "Polynomials," Springer -Verlag, Berlin-Heidelberg, 2010.  Google Scholar

[27]

W. A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 671-694.  Google Scholar

[28]

C. Swan, I. Cummins and R. James, An experimental study of two-dimensional surface water waves propagating on depth-varying currents, J. Fluid Mech., 428 (2001), 273-304. doi: 10.1017/S0022112000002457.  Google Scholar

[29]

G. Thomas and G. Klopman, Wave-current interactions in the nearshore region, in "Gravity Waves in Water of Finite Depth," Advances in Fluid Mechanics, Wessex Institute of Technology, Southhampton (1997), 215-319. Google Scholar

[30]

J.-P. Tignol, "Galois' Theory of Algebraic Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 2001.  Google Scholar

[31]

E. Varvaruca, On some properties of traveling water waves with vorticity, SIAM J. Math. Anal., 39 (2008), 1686-1692. doi: 10.1137/070697513.  Google Scholar

[32]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.  Google Scholar

show all references

References:
[1]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[2]

A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16. doi: 10.1016/j.euromechflu.2010.09.008.  Google Scholar

[3]

A. Constantin, "Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis," CBMS-NSF Series in Applied Mathematics, Vol. 81, SIAM, Philadelphia, 2011. Google Scholar

[4]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.  Google Scholar

[5]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773.  Google Scholar

[6]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[7]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[8]

A. Constantin, J. Escher and H.-C. Hsu, Pressure beneath a solitary water wave: mathematical theory and experiments, Arch. Rational Mech. Anal., 201 (2011), 251-269. doi: 10.1007/s00205-011-0396-0.  Google Scholar

[9]

A. Constantin, D. Sattinger and W. Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech., 548 (2006), 151-163. doi: 10.1017/S0022112005007469.  Google Scholar

[10]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.  Google Scholar

[11]

A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950. doi: 10.1002/cpa.20165.  Google Scholar

[12]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Philos. Trans. Roy. Soc. London A, 365 (2007), 2227-2239. doi: 10.1098/rsta.2007.2004.  Google Scholar

[13]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.  Google Scholar

[14]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Rational Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4.  Google Scholar

[15]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Rational Mech. Anal., 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x.  Google Scholar

[16]

A. F. Teles da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), 281-302. doi: 10.1017/S0022112088002423.  Google Scholar

[17]

M. Ehrnström, On the streamlines and particle paths of gravitational water waves, Nonlinearity, 21 (2008), 1141-1154. doi: 10.1088/0951-7715/21/5/012.  Google Scholar

[18]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 2001.  Google Scholar

[19]

M. Goldshtik and F. Hussain, Inviscid separation in steady planar flows, Fluid Dynamics Research, 23 (1998), 235-266. doi: 10.1016/S0169-5983(98)00017-3.  Google Scholar

[20]

D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves, Philos. Trans. Roy. Soc. London A, 365 (2007), 2241-2251. doi: 10.1098/rsta.2007.2005.  Google Scholar

[21]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows,, Quart. Appl. Math. (in print)., ().   Google Scholar

[22]

I. G. Jonsson, Wave-current interactions, in "The Sea" (eds. B. Le Mehaute and D. M. Hanes), J. Wiley, (1990), 65-120. Google Scholar

[23]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215. doi: 10.1017/S0022112008002371.  Google Scholar

[24]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves, Eur. J. Mech. B Fluids, 27 (2008), 96-109. doi: 10.1016/j.euromechflu.2007.04.004.  Google Scholar

[25]

D. H. Peregrine, Interaction of water waves and currents, Adv. Appl. Mech., 16 (1976), 9-117. doi: 10.1016/S0065-2156(08)70087-5.  Google Scholar

[26]

V. V. Prasolov, "Polynomials," Springer -Verlag, Berlin-Heidelberg, 2010.  Google Scholar

[27]

W. A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 671-694.  Google Scholar

[28]

C. Swan, I. Cummins and R. James, An experimental study of two-dimensional surface water waves propagating on depth-varying currents, J. Fluid Mech., 428 (2001), 273-304. doi: 10.1017/S0022112000002457.  Google Scholar

[29]

G. Thomas and G. Klopman, Wave-current interactions in the nearshore region, in "Gravity Waves in Water of Finite Depth," Advances in Fluid Mechanics, Wessex Institute of Technology, Southhampton (1997), 215-319. Google Scholar

[30]

J.-P. Tignol, "Galois' Theory of Algebraic Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 2001.  Google Scholar

[31]

E. Varvaruca, On some properties of traveling water waves with vorticity, SIAM J. Math. Anal., 39 (2008), 1686-1692. doi: 10.1137/070697513.  Google Scholar

[32]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.  Google Scholar

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